The mathematical theory of frames is used at the Acoustics Research Institute (ARI) of the OeAW to study representations of acoustic signals. However, this fundamental theory is also used in other applications with great success.
The ARI mathematics cluster thus invites OeAW colleague Ronny Ramlau to give an ARI guest talk and share his research on this theory of joint interest with quite different applications.
Ronny Ramlau is Professor and Head of Institute at the Industrial Mathematics Institute at the Johannes Kepler University Linz and Scientific Director of the Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences (OeAW), Linz, Austria.
TALK: "Frame Decompositions - Theory and Applications in Adaptive Optics and Tomography"
ABSTRACT: R. Ramlau and S. Hubmer, Johannes Kepler University Linz, Industrial Mathematics Institute, and Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria:
The theory of Inverse Problems is concerned with the recovery of underlying information from measurements. A prominent example for such a task is the recovery of the density distribution of a human body from Computerized Tomography (CT) data. Mathematically, the relation between data y and the searched-for quantity x can be described by an operator equation Ax = y, i.e., the operator A needs to be inverted to recover x. This is in particular complicated if A is ill-posed and only noisy measurements are available. Regularization methods are commonly used to stabilize the inversion process. They are computationally cheap if the singular value decomposition is available. An alternative is a frame based decomposition of the operator A, in which case also x is represented by a frame. In our talk we present a regularization theory based on frame decompositions and present applications both from Computerized Tomography and Adaptive Optics.